which integral below is equivalent to the given integral, ∫(√(x^5)/3) dx?\n∫(3x^(5/2)) dx\n∫((1/3)x^5)…

Explanation:

Step1: Rewrite the square - root as a fractional exponent

Recall that $\sqrt{x^5}=x^{\frac{5}{2}}$. So the integral $\int(\frac{\sqrt{x^5}}{3})dx=\int(\frac{x^{\frac{5}{2}}}{3})dx$.

Step2: Use the constant - multiple rule of integration

The constant - multiple rule states that $\int cf(x)dx = c\int f(x)dx$ for a constant $c$. Here $c = \frac{1}{3}$, so $\int(\frac{x^{\frac{5}{2}}}{3})dx=\int(\frac{1}{3}x^{\frac{5}{2}})dx$.

Answer:

$\int(\frac{1}{3}x^{\frac{5}{2}})dx$ (the fourth option)